Ok, I have no idea what's going on here.
For the past, well, I don't know... let's say, since I installed Mountain Lion on both my iMac and my MBP, every time I try to connect to one from the other I always get "Connection Failed". I then click on "Connect As" and put in my Registered User credentials. All is well. Well, no, not really. In the past I used to be able to use my Apple ID for this and it would do it automatically. Now, when I click on "Connect As" and choose "Using Apple ID" the box shakes (as if I've entered something wrong). I get similar errors when I try to Share Screen.
I spent over an hour on the phone with Apple Care today, speaking to three different technicians. The last one basically told me that I'm always supposed to use Registered Users and not to worry that using Apple ID doesn't work. In fact, he questioned if it was ever supposed to work like that.
Well, tonight, out of the blue it's working on my MBP when connecting to my iMac! I'll admit I did a lot of stuff but nothing when it started working on it's own. However, like tedus, it only works in one direction. From my iMac to my MBP I am still getting "Connection Failed".